Standard statistical thermodynamic views of temperature fluctuations predict a magnitude ( < (∆T ) 2 >/T ) ≈ (k B /C) for a system with heat capacity C. The extent to which low temperatures can be well defined is discussed for those systems which obey the thermodynamic third law in the form lim (T →0) C = 0. Physical limits on the notion of very low temperatures are exhibited for simple systems. Application of these concepts to bound Bose condensed systems are explored, and the notion of bound Boson superfluidity is discussed in terms of the thermodynamic moment of inertia.

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The resonant transition in the Paul trap is studied by the time-space transformation and function series expansion. A new selection rule for the transition in the trap is derived strictly. The probability in the ground state and some transition rates to the excited states are calculated. The comparison with the former studies shows that our results are more accurate and credible.

The observables of continuous eigenvalues are defined in an infinite-dimensional ket space. The complete set of such eigenstates demands a spectrum density factor, for example, for the photons in the free space and electrons in the vacuum. From the derivation of the Casimir force without an artificial regulator we determine the explicit expression of the spectrum density factor for the photon field to be an exponential function. The undetermined constant in the function is fixed by the experimental data for the Lamb shift. With that, we predict that there exists a correction to the Casimir force.